There are a lot of different things you can do to the end of the drill in this situation.
To do what you describe by hand, a sharp corner is going to grab and possibly break the drill. I suggest using a drill with a neutral rake, minimal clearance and even a chamfer or radius on the corner. This doesn't mean its dull, it means it has different, less aggressive cutting geometry and the cutting edge will be stronger and live longer. Grinding small flats on the leading edge of each flute will create a stronger edge that is far more controllable than a sharp edge - less grabbing and longer life.
If you use OldCarGuy's suggestion, keep in mind the masonary drills create a lot of heat when used for said purpose. It's a good suggestion, though - the carbide is a lot stronger than a typical drill. Masonary drills don't have cutting geometry, though - they basically push material out of the way when used on steel. I assume you have no way to grind carbide otherwise there are a few things you can do to a standard masonary bit to make them cut fairly well.
It's possible that you have work hardened this area of your part so you might be in for a bit of work now. Keep the generated heat to a minimum.
If you are not making chips, you're just making heat.
Scott